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In-Depth Information
R
1
:
IF
x
is
G
,THEN
y
is
1
1
1
F
R
2
:
IF
x
is
2
1
, THEN
y
is
2
1
G
F
:
: :
:
R
M
:
IF
x
is
1
M
, THEN
y
is
1
M
G
F
Now, if
x
is
zc , then the objective is
to determine the corresponding output fuzzy set through the Mamdani rule
inferencing mechanism. The procedure is as follows.
G
c , is given as the input fuzzy set and
1
1
1
GG
1
1
Each fuzzy rule above can be regarded as a fuzzy relation:
> @
l
RX
uo
:
0,1
computed as
P
X
u
YI
P
x
,
P
y
,
l
R
G
l
F
l
1
1
where the operator
I
can be either a
fuzzy implication
or a
conjunction operator
such as a
t-norm
. It is to be noted that
.,
I
is computed on the Cartesian product
space
X
u ,
i.e.
for all possible pairs of
x
and
y
from the domain, using the
Mamdani implication
I
P
x
,
P
y
min
P
x
,
P
y
.
l
l
l
l
G
F
G
F
1
1
1
1
Once the fuzzy relation (
R
) is computed for each rule
l
= 1, 2, 3, ...,
M
, the fuzzy
relation
R
for the entire rule base is computed taking the element-wise maximum
of all (
R
)
i.e. R
is the union of all (
R
), for
l
= 1, 2, 3, ...,
M
. From this fuzzy
relation the output fuzzy set is computed directly by applying a max-min
composition and written as
l
F
c
D .
l
R
G
out
1
1
Using the minimum
t-norm
operator, the max-min composition is obtained as
max
min
P
y
P
x
,
P
x
,
y
l
1
1
R
F
G
c
X
X
,
Y
out
1
The final result of this max-min composition is nothing but the desired output
fuzzy set. The COG of the output fuzzy set gives the equivalent crisp output (
y
coordinates).
The procedure described above can be circumvented by the following few
steps:
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